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If an operator $A$ on a Hilbert space $H$ generates a strongly continuous semigroup, does then the operator $B$ on $H \oplus H$ given by the matrix $$ B := \begin{pmatrix} 0 & \mathrm{id} \\ A & 0\end{pmatrix}$$ generate a semigroup as well? This would then yield a solution to the wave equation $u^{\prime\prime} = A u$ on $H\oplus H$.

If this is not generally true, I would be very interested in a counterexample. Also, if this is not generally true, what properties of $A$ are needed in order for this to be true?

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  • $\begingroup$ The Hille-Yosida theory applies to the operator $B$ as well. See for example Sec.10.3 of Brezis' book Functional Analysis, Sobolev Spaces and Partial Differential Equations $\endgroup$
    – Liviu Nicolaescu
    Commented May 15, 2016 at 23:12
  • $\begingroup$ For nonlinear semigroups see the paper by V. Barbu: Sur un problème aux limites pour une classe d'équations différentielles non linéaires abstraits du deuxième ordre en $t$. (French) C. R. Acad. Sci. Paris Sér. A-B 274 (1972) $\endgroup$
    – Liviu Nicolaescu
    Commented May 15, 2016 at 23:18
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    $\begingroup$ If $B$ is unbounded, then this is never a semigroup generator. You have to work in a different product space, the so-called energy space. $\endgroup$
    – András Bátkai
    Commented May 16, 2016 at 6:26

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You need two additional assumptions: the operator $A$ has to be a so-called cosine function generator, and your product space has to be $V\times H$ with a space $V\subset H$.

Cosine function generator is more than sectorial, it is more or less when the numerical range of $A$ is in a parabola (selfadjoint negative definite is ok, this is discussed in the references Liviu mentioned).

The space $V$ is difficult, but if $A$ is selfadjoint, then it is essentially $D(A^{1/2})$.

See Section VI.3 in

Klaus-Jochen Engel and Rainer Nagel, MR 1721989 One-parameter semigroups for linear evolution equations, ISBN: 0-387-98463-1. (here a downloadable version)

or Section 7.4 in

Haase, Markus The functional calculus for sectorial operators. Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. xiv+392 pp. ISBN: 978-3-7643-7697-0; 3-7643-7697-X

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